Is there a correlation between nationalism/conservatism and lack of
judicial independence/rule of law?
For this purpose, there is not a single nationalism/conservatism ideology, and determining how to classify an ideology gets harder as one gets more removed from a specific historical context. It has a pretty clear answer in the context of particular historical political movements, but is not universal in all times and places.
Many of the nationalist revolutions of the 18th and 19th centuries sought to replace monarchies in which the person of the monarch, rather than a "people" making up a "nation" represented through an institution that is an organization, sought to replace absolutism rooted in the divine right of kings who ruled unbound by the law, with a national government organization subject to the rule of law as enforced by an independent judiciary.
This would be a fair characterization of the American Revolution, several stages of the French Revolutions, most of the first rounds of Latin American wars of independence, the movement for Independence in India from the United Kingdom, the Irish struggle for independence, the first stage of the Russian Revolution, and the Finnish nationalist movements, for example.
It would not be unfair to characterize the efforts of the Southern states in the U.S. Civil War as a conservative nationalist movement, and in that conflict, neither the North nor the South was really opposed to the concepts of an independent judiciary or the rule of law, the sides simply disagreed over what the laws should be, what definition of "nation" should be adopted, and who should participate.
In tbe later stages of the Russian Revolution in 1917, and in the second half of the 20th century independence movements in Africa and Asia, the movements often had significant nationalist components, but also often sought either initially or not long after an initial stage of an independence movement, to adopt one party systems with a communist ideology, abrogating legal rights of the previous capitalist/colonial regimes (see, e.g. Land Reform in Zimbabwe and the Communist Chinese Revolution and the Cambodian regime). These one party states were often authoritarian, did not have independent judiciaries, did not place a premium on rule of law (without being entirely lawless either), and were economically radically liberal rather than being conservative.
Political movements happen in waves. In 21st century Europe, for example, I think it is fair to state that nationalist/conservative political forces seek to weaken the independence of the judiciary and the rule of law. On the other hand, in the Middle East and North Africa, nationalist/conservative political forces are often coalitions of the military, senior civil servants, and former revolutionaries who moderated themselves after victory was secured in wars of independence, who see themselves as political forces resisting religious political movements that seek to make their rules supreme over secular authority without independent courts. But the religious movements are often more socially conservative, seek to reduce judicial independence in concert with a desire to root out pervasive corruption in secular courts, and while not nationalist per se (favoring a larger religious sect identity) attract followers by arguing they they can be more true to religious law than corrupt secular authorities are to secular laws, whose adoption in the legislative process can seem tainted relative to the putatively eternal and unchanging religious laws.
So called "Maoist" revolutionaries in India are another ambiguous case. Basically, they want land reform, which at first blush seems like an anti-rule of law position. But they want it to replace a society in which large land owners who derived their holdings from aristocratic families of the fairly distant colonial and pre-colonial past, and rule their domains like autocrats through the ruse of seemingly secular Western style democratic and political institutions are replaced by a more egalitarian regime in which the powerful are held accountable by the a democratically constituted state, where the masses have effective practical remedies to obtain fair treatment in society, and whether the community and its peoples have primacy over individual notable affluent families. Where does that fit in your taxonomy?
In much of modern Africa and Europe, nationalist movements often seek to dismember states that are either multi-ethnic, or have arbitrary boundaries that don't reflect the underlying ethnic distributions of the people in those countries, often because the boundaries were drawn arbitrarily by colonial powers. To the extent that they want big changes, they can't be truly called "conservative" even if the backers (e.g. of Northern Italian independence, Basque independence, Catalan independence) are economically affluent anti-socialist subcultures within the society where they are part of a state. But, in Japan or Korea, for example, nationalism is a movement seeking to prevent the dilution of an extreme ethnically homogeneous nation-state who strongly support the existing state and are conservative in a more traditional sense, but perhaps not as instinctively "anti-socialist" economically.